Transition temperature of the weakly interacting Bose gas: perturbative solution of the crossover equations in the canonical ensemble

We compute the shift of the critical temperature T-c with respect to the id eal case for a weakly interacting uniform Bose gas. We work in the framewor k of the canonical ensemble, extending the criterion of condensation provid ed by the canonical particle counting statistics for the zero-momentum stat e of the uniform ideal gas. The perturbative solution of the crossover equa tion to lowest order in powers of the scattering length yields (T-c - T-c(0 ))/T-c(0) = -0.93a rho (1/3), where T-c(0) is the transition temperature of the corresponding ideal Bose gas, a is the scattering length, and rho is t he particle number density. This result is at variance with the standard gr and canonical prediction of a null shift of the critical temperature in the lowest perturbative order. The non-equivalence of statistical ensembles fo r the ideal Bose gas is thus confirmed (at the lowest perturbative level) a lso in the presence of interactions.